# Second Moment Convergence Rates for Uniform Empirical Processes

- You-You Chen
^{1}Email author and - Li-Xin Zhang
^{1}

**2010**:972324

https://doi.org/10.1155/2010/972324

© Y.-Y. Chen and L.-X. Zhang. 2010

**Received: **21 May 2010

**Accepted: **19 August 2010

**Published: **23 August 2010

## Abstract

## Keywords

## 1. Introduction and Main Results

if and . The converse part was proved by the study of Erdös in [2]. Obviously, the sum in (1.1) tends to infinity as . Many authors studied the exact rates in terms of (cf. [3–5]). Chow [6] studied the complete convergence of , . Recently, Liu and Lin [7] introduced a new kind of complete moment convergence which is interesting, and got the precise rate of it as follows.

Theorem A.

holds, if and only if , , and .

Other than partial sums, many authors investigated precise rates in some different cases, such as *U*-statistics (cf. [8, 9]) and self-normalized sums (cf. [10, 11]). Zhang and Yang [12] extended the precise asymptotic results to the uniform empirical process. We suppose
is the sample of
random variables and
is the empirical distribution function of it. Denote the uniform empirical process by
,
, and the norm of a function
on
by
. Let
,
be the Brownian bridge. We present one result of Zhang and Yang [12] as follows.

Theorem B.

Inspired by the above conclusions, we consider second moment convergence rates for the uniform empirical process in the law of iterated logarithm and the law of the logarithm. Throughout this paper, let denote a positive constant whose values can be different from one place to another. will denote the largest integer . The following two theorems are our main results.

Theorem 1.1.

Theorem 1.2.

Remark 1.3.

Consequently, explicit results of (1.4) and (1.5) can be calculated further.

## 2. The Proofs

In order to prove Theorem 1.1, we present several propositions first.

Proposition 2.1.

Proof.

Proposition 2.2.

Proof.

From (2.6), (2.7), and (2.9), we get . Proposition 2.2 has been proved.

Proposition 2.3.

Proof.

The calculation here is analogous to (2.1), so it is omitted here.

Proposition 2.4.

Proof.

Put the three parts together, we get that uniformly in as . Using Toeplitz's lemma again, we have

Therefore, we get . So far, we only need to prove . Use the inequality again and follow the proof of , we can get this result. The proof of the proposition is completed now.

Proof of Theorem 1.1.

Proof of Theorem 1.2.

Combine (2.22), (2.23), and (2.24)together, we get the result of Theorem 1.2.

## Declarations

### Acknowledgment

This work was supported by NSFC (No. 10771192) and ZJNSF (No. J20091364).

## Authors’ Affiliations

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